Help me matlab,find $f(x)$ from $f(g(x))$....thanks a lot!

Question

How to find $f(x)$ from $f(g(x))$, $g(x)$, ex:

$g(x)=1-x^2$

$f(g(x))=(x^4+x^2)/(1+x^2)$

$f(1/2)=$

use matlab code....I don't know..help me thanks a lot!

Answer 1

For $g(x) = 1 - x^2$ to be equal to $1/2$, $x = 1/\sqrt{2}$.

So: $$f(1/2) = \frac{1/4 + 1/2}{1 + 1/2} = 0.75/1.5 = 1/2$$

Answer 2

I give you an example, and hope you can translate it to Matlab. I don't know Matlab much but I think you can do it by yourself.

Assume you have $$h(x)=f(g(x))=4x^2+4x+7$$ and $$g(x)=2x+1$$ and want to find $f(x)$. Define $h(x)$ and $g(x)$ in Matlab environment and solve $$g(x)=t$$ respect to $x$(which $t$ is a parameter) You get: $$x=0.5t-0.5$$ Now put what you got into $h(x)$, so you get: $$h(0.5t-0.5)=t^2+6$$ This is your desired function $f(x)$ if you replace $t$ to $x$.